Does there exist a battery-powered, programmable, millivolt thermostat

I'm going to go ahead and say that YES, you can do as you propose, wiring the two thermostats in parallel.

While, as others have correctly pointed out, this will not let you achieve temperature control in both the upstairs and downstairs simultaneously, I'm going to take you at your word that you simply do not want it to be uncomfortably cold in the basement when occupying the basement, and infer that you don't care what happens upstairs during those times.

In this case, go for it--install the second thermostat.

The upstairs area of the house will in fact be over-heated when you use this basement thermostat, but you may still consider the result to be an improvement over the current situation. Just remember to reset the basement thermostat to cooler temperature before you leave to go back upstairs or you will find yourself having to make the trip back down later.

The "correct" way to solve the problem is to split your system into two zones, which requires installing baffles/valves on your heating ducts/pipes, which would allow independent control of upstairs and downstairs.

If you are using traditional thermostats, then adding a thermostat in parallel will change the heat anticipator function in both of them. This is a function that reduces furnace run times to improve temperature stability. If both are always connected, lower the HA settings in each to fix.

You'll need a transformer to step down mains voltage, to 24 volts that the thermostat can use. And you'll need a relay that can work with 24 volts.

You'll wire it up something like this.

When the thermostat calls for heat, the relay contact will be pulled closed. This will signal the fireplace to turn on.

_{NOTE: the three contacts used on the thermostat are R, W, and C. In the drawing the C may or may not look like a C, so I just wanted to make that clear.}